Random stuff/Open Thread

i. A lecture on mathematical proofs:

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ii. “In the fall of 1944, only seven percent of all bombs dropped by the Eighth Air Force hit within 1,000 feet of their aim point.”

From wikipedia’s article on Strategic bombing during WW2. The article has a lot of stuff. The ‘RAF estimates of destruction of “built up areas” of major German cities’ numbers in the article made my head spin – they didn’t bomb the Germans back to the stone age, but they sure tried. Here’s another observation from the article:

“After the war, the U.S. Strategic Bombing Survey reviewed the available casualty records in Germany, and concluded that official German statistics of casualties from air attack had been too low. The survey estimated that at a minimum 305,000 were killed in German cities due to bombing and estimated a minimum of 780,000 wounded. Roughly 7,500,000 German civilians were also rendered homeless.” (The German population at the time was roughly 70 million).

Although over 21,000 American soldiers were given varying sentences for desertion during World War II, including 49 death sentences, Slovik’s was the only death sentence that was actually carried out.[1][3][4]

During World War II, 1.7 million courts-martial were held, representing one third of all criminal cases tried in the United States during the same period. Most of the cases were minor, as were the sentences.[2] Nevertheless, a clemency board, appointed by the Secretary of War in the summer of 1945, reviewed all general courts-martial where the accused was still in confinement.[2][5] That Board remitted or reduced the sentence in 85 percent of the 27,000 serious cases reviewed.[2] The death penalty was rarely imposed, and those cases typically were for rapes or murders. […] In France during World War I from 1917 to 1918, the United States Army executed 35 of its own soldiers, but all were convicted of rape and/or unprovoked murder of civilians and not for military offenses.[13] During World War II in all theaters of the war, the United States military executed 102 of its own soldiers for rape and/or unprovoked murder of civilians, but only Slovik was executed for the military offense of desertion.[2][14] […] of the 2,864 army personnel tried for desertion for the period January 1942 through June 1948, 49 were convicted and sentenced to death, and 48 of those sentences were voided by higher authority.”

What motivated me to read the article was mostly curiosity about how many people were actually executed for deserting during the war, a question I’d never encountered any answers to previously. The US number turned out to be, well, let’s just say it’s lower than I’d expected it would be. American soldiers who chose to desert during the war seem to have had much, much better chances of surviving the war than had soldiers who did not. Slovik was not a lucky man. On a related note, given numbers like these I’m really surprised desertion rates were not much higher than they were; presumably community norms (”desertion = disgrace’, which would probably rub off on other family members…’) played a key role here.

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iv. Chess and infinity. I haven’t posted this link before even though the thread is a few months old, and I figured that given that I just had a conversation on related matters in the comment section of SCC (here’s a link) I might as well repost some of this stuff here. Some key points from the thread (I had to make slight formatting changes to the quotes because wordpress had trouble displaying some of the numbers, but the content is unchanged):

u/TheBB:
“Shannon has estimated the number of possible legal positions to be about 1043. The number of legal games is quite a bit higher, estimated by Littlewood and Hardy to be around 1010^5 (commonly cited as 1010^50 perhaps due to a misprint). This number is so large that it can’t really be compared with anything that is not combinatorial in nature. It is far larger than the number of subatomic particles in the observable universe, let alone stars in the Milky Way galaxy.

As for your bonus question, a typical chess game today lasts about 40­ to 60 moves (let’s say 50). Let us say that there are 4 reasonable candidate moves in any given position. I suspect this is probably an underestimate if anything, but let’s roll with it. That gives us about 42×50 ≈ 1060 games that might reasonably be played by good human players. If there are 6 candidate moves, we get around 1077, which is in the neighbourhood of the number of particles in the observable universe.”

u/Wondersnite:
“To put 1010^5 into perspective:

There are 1080 protons in the Universe. Now imagine inside each proton, we had a whole entire Universe. Now imagine again that inside each proton inside each Universe inside each proton, you had another Universe. If you count up all the protons, you get (1080 )3 = 10240, which is nowhere near the number we’re looking for.

You have to have Universes inside protons all the way down to 1250 steps to get the number of legal chess games that are estimated to exist. […]

Imagine that every single subatomic particle in the entire observable universe was a supercomputer that analysed a possible game in a single Planck unit of time (10-43 seconds, the time it takes light in a vacuum to travel 10-20 times the width of a proton), and that every single subatomic particle computer was running from the beginning of time up until the heat death of the Universe, 101000 years ≈ 1011 × 101000 seconds from now.

Even in these ridiculously favorable conditions, we’d only be able to calculate

1080 × 1043 × 1011 × 101000 = 101134

possible games. Again, this doesn’t even come close to 1010^5 = 10100000 .

Basically, if we ever solve the game of chess, it definitely won’t be through brute force.”

About me/this blog

This blog is mainly a site where I keep track of and share some of the stuff I read and learn. Only a small subset of the posts on this blog deal with economics – I have diverse interests, and as the category cloud in the sidebar below illustrates this blog contains posts about all kinds of stuff: Mathematics, physics, statistics, geology, geography, health care and medicine, psychology, evolutionary biology, genetics, history, anthropology, archaeology, chess, …

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